TITLE: Calculate $2^{1500}$
QUESTION [5 upvotes]: Is there a way of approximating $2^{1500}$ using pen and paper? I just took $2^{1500}$ as an example but it can be $2^{150000}$ or $12^{95}$. I am just looking for pen and paper method.
By approximate, I mean a solution like $3.5*10^{21}$ or something along those lines.
I was thinking of doing the following: $2^{1500} =x$ then take the $log$ of both sides but I don't know if that makes sense.

REPLY [16 votes]: If one only need a quick and very rough estimate, one should find 
a number close to what you need to estimate and yet you know how to handle.
You then expand the result from the number you know.
For the case $2^{1500}$, one can use the fact $2^{10} = 1024$ is very close to a power of $10$ and $\log 2 \sim 0.7$. We have
$$\begin{align}
2^{1500} = & (2^{10})^{150} = (1.024 \times 10^3)^{150}\\
\sim & e^{0.024\times 150} \times 10^{450} = e^{3.6} \times 10^{450} = e^{5\times 0.7 + 0.1} \times 10^{450}\\
\sim & 2^5 e^{0.1} \times 10^{450} = 3.52 \times 10^{451}
\end{align}$$
Compare with the correct answer $\sim 3.50746 \times 10^{451}$, this estimate is accurate up to around $1\%$. Not bad for a calculation you can do it in your head.